529 research outputs found
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials
Generalizing recent results of Egge and Mongelli, we show that each diagonal
sequence of the Jacobi-Stirling numbers \js(n,k;z) and \JS(n,k;z) is a
P\'olya frequency sequence if and only if and study the
-total positivity properties of these numbers. Moreover, the polynomial
sequences \biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and}
\quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0} are proved to be
strongly -log-convex. In the same vein, we extend a recent result of
Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan
polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising
from the Lambert function, we obtain a neat proof of the unimodality of the
latter sequence, which was proved previously by Kalugin and Jeffrey.Comment: 17 pages, 2 tables, the proof of Lemma 3.3 is corrected, final
version to appear in Advances in Applied Mathematic
Examples and counterexamples in Ehrhart theory
This article provides a comprehensive exposition about inequalities that the
coefficients of Ehrhart polynomials and -polynomials satisfy under various
assumptions. We pay particular attention to the properties of Ehrhart
positivity as well as unimodality, log-concavity and real-rootedness for
-polynomials.
We survey inequalities that arise when the polytope has different normality
properties. We include statements previously unknown in the Ehrhart theory
setting, as well as some original contributions in this topic. We address
numerous variations of the conjecture asserting that IDP polytopes have a
unimodal -polynomial, and construct concrete examples that show that these
variations of the conjecture are false. Explicit emphasis is put on polytopes
arising within algebraic combinatorics.
Furthermore, we describe and construct polytopes having pathological
properties on their Ehrhart coefficients and roots, and we indicate for the
first time a connection between the notions of Ehrhart positivity and
-real-rootedness. We investigate the log-concavity of the sequence of
evaluations of an Ehrhart polynomial at the non-negative integers. We
conjecture that IDP polytopes have a log-concave Ehrhart series. Many
additional problems and challenges are proposed.Comment: Comments welcome
h-Vectors of matroids and logarithmic concavity
Let M be a matroid on E, representable over a field of characteristic zero.
We show that h-vectors of the following simplicial complexes are log-concave:
1. The matroid complex of independent subsets of E. 2. The broken circuit
complex of relative to an ordering of E. The first implies a conjecture of
Colbourn on the reliability polynomial of a graph, and the second implies a
conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based
on the geometric formula for the characteristic polynomial of Denham,
Garrousian, and Schulze.Comment: Major revision, 9 page
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